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Understanding Image Features: Edges, Corners, and Convolution Techniques

This document provides an overview of image features, specifically focusing on edges and corners, and outlines essential properties of convolution and linear systems. It covers fundamental concepts such as linearity, superposition, and neighborhood filtering techniques, including median filtering for noise reduction. The document delves into edge detection methods, particularly the Canny, Sobel, and Prewitt detectors using MATLAB, and discusses the mathematical underpinnings of corner detection. This is aimed at students and professionals interested in computer vision and image processing.

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Understanding Image Features: Edges, Corners, and Convolution Techniques

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  1. Image Features - I Hao Jiang Computer Science Department Sept. 22, 2009

  2. Outline • Summary of convolution and linear systems • Image features • Edges • Corners • Programming Corner Detection

  3. Properties of Convolution 1. Commutative: f * g = g * f 2. Associative (f * g) * h = f *(g * h) 3. Superposition (f + g) * h = f * h + g * h N M full (N+M-1)x(N+M-1)

  4. Linear System f h g = f * h Linear: a f1+ b f2 => a g1 + b g2 where the response of f1 is g1 and the response of f2 is g2 Shift invariant: if f => g, then f(n-m) => g(n-m)

  5. Composite Linear System f h1 h2 h1 + h2 f h1 h2 h1*h2

  6. Nonlinear Filtering • Neighborhood filtering can be nonlinear • Median Filtering • 1 1 • 1 2 1 • 1 1 1 Mask [1 1 1 ] • 1 1 • 1 1 1 • 1 1 1

  7. Median Filtering in Denoising Original Image Add 10% pepper noise

  8. Median Filtering for Denoising Median filter with 3x3 square structure element

  9. Median Filtering for Denoising Median filter with 5x5 square structure element

  10. Compared with Gaussian Filtering Kernel size 5x5 and sigma 3 Kernel size 11x11 and sigma 5

  11. Image Features

  12. Image Local Structures Step Ridge Valley Peak Corner Junction

  13. Image Local Structures Line Structures: “Edge” Step Ridge Valley Point Structures: “Corners” Peak Corner Junction

  14. Regions

  15. An Example edge Region corners

  16. Edge Detection in Matlab >> im = imread('flower.jpg'); >> im = im2double(im); >> im = rgb2gray(im); >> ed = edge(im, 'canny', 0.15);

  17. How to Find an Edge? A 1D edge

  18. f(x) f’(x) f’’(x)

  19. Extend to 2D b There is a direction in which image f(x,y) increases the fastest. The direction is called the gradient direction. Gradient [df/dx df/dy] Magnitude: sqrt(fx^2 + fy^2) Direction: atan2(fy, fx) a

  20. Finite Difference • Approximating derivatives using finite difference. • Finite difference and convolution

  21. Noise Reduction 0.01 noise 0.03 noise

  22. Noise Reduction

  23. Gaussian Filtering in Edge Detection

  24. Gaussian Filtering in Edge Detection image h * (g * f) = (h * g) * f Difference of Gaussian Kernel Difference Kernel Gaussian Kernel

  25. Edge Detection in Images • Gaussian smoothed filtering in x and y directions: Ix, Iy • Non-maximum suppression for |Ix|+|Iy| • Edge Tracing – double thresholding.

  26. Edge Detection Using Matlab • Canny edge detector: edge(image, ‘canny’, threshold) • Sobel edge detector: edge(image, ‘sobel’, threshold) • Prewitt edge detector: edge(image, ‘prewitt’, threshold)

  27. Berkeley Segmentation DataSet [BSDS] D. Martin, C. Fowlkes, D. Tal, J. Malik. "A Database of Human Segmented Natural Images and its Application to Evaluating Segmentation Algorithms and Measuring Ecological Statistics”, ICCV, 2001

  28. Corner Detection • Corner is a point feature that has large changing rate in all directions. Peak Step Line Flat region

  29. Find a Corner Compute matrix H = Ix2 Ixy Ixy Iy2 = in each window. If the ratio (Ix2 * Iy2 – Ixy ^2 ) ------------------------ > T (Ix2 + Iy2 + eps) We have a corner

  30. Corner Detection Programming

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